Understand Solving Equations and Inequalities - 6th Grade Math

1. Which value in the set $\{1,2,3,4\}$ makes the equation $10-x=6$ true?

   $4$. Since $10-4=6$, only $x=4$ satisfies the equation.

2. Which value in the set $\{0, 2, 4\}$ makes $x^2 = 4$ true?

   $2$. Only $2^2 = 4$ is true; $0^2 = 0$ and $4^2 = 16$.

3. Which value in the set $\{2,3,4\}$ makes the equation $\frac{x}{2}=2$ true?

   $4$. Since $\frac{4}{2}=2$, only $x=4$ satisfies the equation.

4. Which values in the set $\{0, 1, 2, 3\}$ make $x + 2 \ge 4$ true?

   $\{2, 3\}$. Test: $2 + 2 = 4 ge 4$ and $3 + 2 = 5 ge 4$ are true.

5. Which value in the set $\{2, 4, 6\}$ makes $x + 1 = 5$ true?

   $4$. Substitute: $4 + 1 = 5$ is true.

6. Identify whether $x=-2$ is a solution of $x+2<0$.

   No, $x=-2$ is not a solution. $-2+2=0$, which is not less than $0$.

7. Which values are you allowed to test when a set is specified?

   Only values from the specified set. You must choose from the given set, not any arbitrary values.

8. Identify whether $x = 5$ is a solution of $2x + 1 = 11$.

   Yes, because $2(5) + 1 = 11$. Substitution confirms the equation is satisfied.

9. Identify whether $x=4$ is a solution of $2x+1=9$.

   Yes, $x=4$ is a solution. $2(4)+1=8+1=9$, which equals the right side.

10. Which values in the set $\{1,2,3,4\}$ make the inequality $2x \ge 6$ true?

    $3$ and $4$. Since $2(3)=6$ and $2(4)=8$, both are $\ge 6$.

11. Identify whether $x = -2$ is a solution of $x + 5 = 3$.

    Yes, because $-2 + 5 = 3$. Substitution confirms the equation is satisfied.

12. Identify whether $x=0$ is a solution of $7x-4=3$.

    No, $x=0$ is not a solution. $7(0)-4=0-4=-4$, which doesn't equal $3$.

13. Which values in $\{0,1,2,3\}$ satisfy $\frac{x}{2} \geq 1$?

    $x=2,3$. $\frac{2}{2}=1$ and $\frac{3}{2}=1.5$ are both $\geq 1$.

14. Which value in $\{2,3,4,5\}$ makes $\frac{x}{5}=\frac{4}{5}$ true?

    $4$. $\frac{4}{5}=\frac{4}{5}$, so $x=4$ works.

15. Which value in the set $\{3,4,5\}$ makes the equation $\frac{x}{5}=1$ true?

    $5$. Since $\frac{5}{5}=1$, only $x=5$ satisfies the equation.

16. Which values in $\{0,2,4,6\}$ satisfy $2x\le 8$?

    $\{0,2,4\}$. $2(0)=0$, $2(2)=4$, $2(4)=8$ all satisfy $le 8$.

17. What is substitution when checking a possible solution?

    Replacing the variable with a given value to test truth. We replace the variable with a number to check if the statement is true.

18. Which values in $\{-1,0,1,2\}$ satisfy $x\ge 1$?

    $\{1,2\}$. Values greater than or equal to 1 are 1 and 2.

19. Which values in $\{-2,-1,0,1,2\}$ make $x^2=4$ true?

    $x=-2$ and $x=2$. Both $(-2)^2=4$ and $2^2=4$ are true.

20. Which value in $\{0,2,4,6\}$ makes $3x=12$ true?

    $4$. $3(4)=12$, so $x=4$ satisfies the equation.

21. Which values in the set $\{-1, 0, 1, 2\}$ make $x - 1 < 1$ true?

    $\{-1, 0, 1\}$. Test each: $-1 - 1 = -2 < 1$, $0 - 1 = -1 < 1$, $1 - 1 = 0 < 1$.

22. Which value in the set $\{1,2,3,4\}$ makes the equation $2(x+1)=10$ true?

    $4$. Since $2(4+1)=2(5)=10$, only $x=4$ satisfies.

23. Which values in $\{1,2,3,4,5\}$ satisfy $x+2>6$?

    $\{5\}$. Only $5+2=7>6$, others give $le 6$.

24. Which value in the set $\{2, 3, 4\}$ makes $\frac{x}{3} > 1$ true?

    $4$. Only $\frac{4}{3} = 1.33... > 1$ is true.

25. What is substitution when checking a solution to $x + 7 = 12$?

    Replace $x$ with the given number and evaluate both sides. This process checks if the equation is satisfied.

26. Which value in the set $\{1, 2, 3\}$ makes $5 - x = 2$ true?

    $3$. Substitute: $5 - 3 = 2$ is true.

27. Which value in the set $\{-1,0,1,2\}$ makes the equation $5-x=4$ true?

    $1$. Since $5-1=4$, only $x=1$ satisfies the equation.

28. Which value in the set $\{1,2,3\}$ makes the inequality $x+2>4$ true?

    $3$. Since $3+2=5>4$, only $x=3$ satisfies the inequality.

29. What does it mean for a value to be a solution of an inequality?

    A value that makes the inequality true when substituted. Substitution verifies if the inequality holds true.

30. Which value in the set $\{1,2,3\}$ makes the equation $2x+1=7$ true?

    $3$. Since $2(3)+1=7$, only $x=3$ satisfies the equation.

31. Which value in $\{1,2,3,4\}$ makes $x+5=8$ true?

    $3$. $3+5=8$, so $x=3$ satisfies the equation.

32. Identify whether $x=-3$ is a solution of $x+2<0$.

    Yes, $x=-3$ is a solution. $-3+2=-1$, which is less than $0$.

33. Which value in the set $\{0, 1, 2\}$ makes $3x = 6$ true?

    $2$. Substitute: $3(2) = 6$ is true.

34. Identify whether $x=-1$ is a solution of $3x+2=-1$.

    Yes, $x=-1$ is a solution. $3(-1)+2=-3+2=-1$, which equals the right side.

35. Which value in the set $\{0,1,2,3\}$ makes the equation $x^2=4$ true?

    $2$. Since $2^2=4$, only $x=2$ satisfies the equation.

36. Which values in the set $\{0,1,2,3\}$ make the inequality $x\le 2$ true?

    $0,1,2$. Values $0$, $1$, and $2$ are all less than or equal to $2$.

37. Find the value in $\{1, 4, 7\}$ that makes $3x - 2 = 19$ true.

    $7$. Substitute: $3(7) - 2 = 21 - 2 = 19$ is true.

38. Which values in the set $\{1, 2, 3, 4\}$ make $10 - 2x > 3$ true?

    $\{1, 2, 3\}$. Test: $10 - 2(1) = 8 > 3$, $10 - 2(2) = 6 > 3$, $10 - 2(3) = 4 > 3$.

39. Which value in the set $\{0,1,2,3\}$ makes the equation $4(x-1)=8$ true?

    $3$. Since $4(3-1)=4(2)=8$, only $x=3$ satisfies.

40. Which value in the set \{0,1,2\} makes the equation $3x=6$ true?

    $2$. Since $3(2)=6$, only $x=2$ satisfies the equation.

41. What is substitution when checking a possible solution for $x$?

    Replace $x$ with the given value and evaluate both sides. Substitution means plugging in a value for the variable.

42. Which statement correctly describes checking $x=2$ in $x+1=4$?

    Substitute: $2+1=3$, so $x=2$ is not a solution. Shows substitution yields $3 eq 4$, proving it's not a solution.

43. Which values are considered when the problem gives a specified set $S$?

    Only values in $S$ can be solutions. Solutions must come from the given set.

44. Which values in the set $\{0,1,2,3\}$ make the inequality $3-x\ge 2$ true?

    $0$ and $1$. Since $3-0=3ge 2$ and $3-1=2ge 2$, both satisfy.

45. Which values in $\{1,2,3,4\}$ satisfy $x+3\leq 6$?

    $x=1,2,3$. $1+3=4$, $2+3=5$, $3+3=6$ are all $\leq 6$; $4+3=7$ is not.

46. Identify whether $x=5$ is a solution of $2x+1=9$.

    No, $x=5$ is not a solution. $2(5)+1=10+1=11$, which doesn't equal $9$.

47. What does it mean for a value to be a solution of an equation?

    A value that makes the equation true when substituted. Substitution verifies if the equation holds true.

48. What is the first step to check whether $x=4$ is a solution of $2x+1=9$?

    Substitute $4$ for $x$ in $2x+1$ and compare to $9$. Substitution checks if the value satisfies the equation.

49. Which values in $\{0,1,2,3,4\}$ satisfy $x<3$?

    $\{0,1,2\}$. Values less than 3 are 0, 1, and 2.

50. Which values in $\{-1,0,1,2\}$ satisfy $2x-1>1$?

    $x=2$ only. Only $2(2)-1=3$ is greater than $1$.

51. Which values in $\{-3,-2,-1,0,1\}$ satisfy $x-1\le -2$?

    $\{-3,-2,-1\}$. $-3-1=-4$, $-2-1=-3$, $-1-1=-2$ all $le -2$.

52. Which value in $\{1,2,3,4\}$ makes $3(x-1)=6$ true?

    $3$. $3(3-1)=3(2)=6$, so $x=3$ works.

53. Which values in the set $\{-2, -1, 0, 1, 2\}$ make $x^2 \le 1$ true?

    $\{-1, 0, 1\}$. Test: $(-1)^2 = 1$, $0^2 = 0$, $1^2 = 1$ all $le 1$.

54. Which value in $\{1,2,3\}$ makes $2x+3=2x+4$ true?

    None. $2x+3=2x+4$ simplifies to $3=4$, which is false.

55. Which value in $\{2,3,4\}$ makes $2x-1=7$ true?

    $4$. $2(4)-1=8-1=7$, so $x=4$ works.

56. Which value in $\{2,4,6\}$ makes $\frac{x}{3}=2$ true?

    $x=6$. $\frac{6}{3}=2$, which equals the right side.

57. Which values in $\{0,1,2,3\}$ make $x^2=4$ true?

    $x=2$ only. Only $2^2=4$; the other values squared don't equal $4$.

58. What is the first step to check whether $x=4$ satisfies $2x+1=9$?

    Substitute $4$ for $x$ in $2x+1=9$. Replace $x$ with $4$ to evaluate the left side of the equation.

59. What does it mean for a value to be a solution of the equation $x+3=10$?

    It makes the equation true when substituted for $x$. A solution satisfies the equation when plugged in for the variable.

60. Which value in $\{1,2,3,4\}$ makes $5=2x+1$ true?

    $2$. $2(2)+1=5$, so $x=2$ satisfies the equation.

61. Which value in $\{1,2,3\}$ makes $5-x=2$ true?

    $x=3$. $5-3=2$, which equals the right side.

62. Which values in the set $\{1, 2, 3, 4\}$ make $2x \le 6$ true?

    $\{1, 2, 3\}$. Test each: $2(1) = 2 le 6$, $2(2) = 4 le 6$, $2(3) = 6 le 6$.

63. Which value in the set $\{1, 2, 3\}$ makes $\frac{x}{2} = 1$ true?

    $2$. Substitute: $\frac{2}{2} = 1$ is true.

64. Which value in $\{0,1,2,3\}$ makes $4+2x=10$ true?

    $3$. $4+2(3)=4+6=10$, so $x=3$ satisfies it.

65. Which values in the set $\{-2,-1,0,1\}$ make the inequality $x<0$ true?

    $-2$ and $-1$. Only negative values $-2$ and $-1$ are less than $0$.

66. Identify whether $x=1$ is a solution of $\frac{x+1}{2}=1$.

    Yes, $x=1$ is a solution. $\frac{1+1}{2}=\frac{2}{2}=1$, which equals the right side.

67. Which value in $\{0,1,2,3\}$ makes $\frac{x}{2}=\frac{3}{2}$ true?

    $3$. $\frac{3}{2}=\frac{3}{2}$, so $x=3$ satisfies it.

68. Which value in the set $\{2,3,4\}$ makes the equation $x+5=8$ true?

    $3$. Since $3+5=8$, only $x=3$ satisfies the equation.

Study these flashcards